Hence, a general strategy to construct Haag-Kastler nets is first to construct Borchers triples
and then to check cyclicity of the vacuum. This program has been first completed in [1]}.
Note that the latter condition is actually very hard to check directly. In [1]}, Lechner
proved instead the so-called modular nuclearity [3]},
which is a sufficient condition for the cyclicity of the vacuum. In this paper, we will provide
new examples of Borchers triples for which the cyclicity of the vacuum can be proved without relying
on modular nuclearity (see Section REF ). We say that a Borchers triple is strictly local
if \(\Omega \)  is cyclic for \({\mathcal {A}}(O)\)  constructed as above. A strictly local Borchers triple
corresponds to a Haag-Kastler net.
